Question

The average distance between base pairs measured parallel to the axis of a DNA molecule is $3.4\text{AA}$. The average molar mass of a pair of nucleotides is $650\mathrm{g}/\mathrm{mol}$. Estimate the length in centimeters of a DNA molecule of molar mass $5.0\times {10}^9\mathrm{g}/\mathrm{mol}$. Roughly how many base pairs are contained in this molecule?

Short answer

The DNA length is approximately 0.26 cm, with about 7.7 million base pairs.

Step-by-step solution

Calculate the Number of Base Pairs

First, determine how many base pairs there are in the DNA molecule with the given molar mass. Use the average molar mass of a pair of nucleotides: $$\text{Number of Base Pairs}=\frac{\text{Molar Mass of DNA}}{\text{Molar Mass per Base Pair}}=\frac{5.0\times {10}^9\mathrm{g}/\mathrm{mol}}{650\mathrm{g}/\mathrm{mol}}=7.6923\times {10}^6.$$ Approximately, this is 7.7 million base pairs.

Estimate the Length of DNA in Angstroms

Use the number of base pairs calculated in the previous step and the average distance between them to estimate the total length in angstroms:$$\text{Length of DNA in Angstroms}=7.7\times {10}^6\times 3.4\text{AA}=2.61846\times {10}^7\text{AA}.$$

Convert Length from Angstroms to Centimeters

Convert the total length of DNA from angstroms to centimeters, knowing that 1 angstrom $(\text{AA})$ is equal to ${10}^{-8}$ centimeters:$$\text{Length in centimeters}=2.61846\times {10}^7\times {10}^{-8}=0.261846\text{ cm}.$$

Key concepts

Base Pairs Calculation

DNA is composed of long chains of nucleotides, paired together as base pairs. To calculate the number of base pairs in a DNA molecule, you need two key pieces of information: the molar mass of the DNA and the average molar mass of a single pair of nucleotides. Each nucleotide pair has an average molar mass of 650 grams per mole. So, for a DNA molecule with a molar mass of $5.0\times {10}^9$ grams per mole, you calculate the number of base pairs by dividing the total molar mass of the DNA by the molar mass per base pair.

Using the formula:

• Base pairs = $\frac{5.0\times {10}^9}{650}$
• Base pairs = $7.6923\times {10}^6$

This results in approximately 7.7 million base pairs for the given DNA molecule, indicating its complexity and length.

Molar Mass of Nucleotides

The molar mass of nucleotides is a crucial factor when estimating the number of base pairs in a DNA strand. Nucleotides are the building blocks of DNA and RNA, consisting of a sugar, a phosphate group, and a nitrogenous base. In DNA, these form the rungs of the helical ladder-like structure.
- Knowing the average molar mass of a nucleotide pair, which is 650 grams per mole, allows us to perform calculations for entire DNA strands by scaling up from the mass of individual nucleotides.
- The total molar mass given for a DNA molecule, such as $5.0\times {10}^9$ grams per mole, helps you calculate whole numbers of base pairs by simple division.

This concept demonstrates how molecular biology and chemistry principles interconnect to decipher DNA's structural characteristics.

Angstrom to Centimeter Conversion

When working with tiny biological structures like DNA, it’s essential to convert measurements to appropriate units. A DNA molecule's length is often measured in angstroms, a convenient unit at atomic scalessince 1 angstrom $(\text{AA})$ is a mere ${10}^{-10}$ meters, or ${10}^{-8}$ centimeters.

To find the length in centimeters:

• First, calculate the total length in angstroms using the base pairs count and the average distance between base pairs.
• For instance, $7.7\times {10}^6$ base pairs with an average distance of $3.4\text{AA}$ yields a total length of $2.61846\times {10}^7$ angstroms.

To convert this length to centimeters, multiply by the conversion factor ${10}^{-8}$:

• So, $2.61846\times {10}^7\times {10}^{-8}=0.261846$ cm.

Understanding these conversions is key to moving between molecular scales and everyday measurements.

Average Distance Between Base Pairs

In the double helix structure of DNA, base pairs are stacked next to each other along the axis. The average distance between each pair is known to be $3.4\text{AA}$, a small but significant measurement that impacts the overall length of the DNA molecule.

This spacing is consistent, meaning that even across millions of base pairs, you can estimate lengths accurately:

• Multiply the distance of $3.4\text{AA}$ by the total base pairs, calculated previously as $7.7\times {10}^6$.
• This results in a total length of $2.61846\times {10}^7$ angstroms for the molecule.

Knowing this average distance is essential for estimating the physical dimensions of DNA strands and understanding the compact nature of genetic material within cells.